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- Consider f(x) = e^x.
Give the Newton form of the polynomial interpolating
through the points (1,e^1), (10,e^(10)), (100,e^(100)).
- What is the relationship between the divided differences
and derivatives? (Hint: think about Taylor.)
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| /20 |
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- Consider the quotient
2 2
12 + 6 b x + b x
q = ------------------
2 2
12 - 6 b x + b x
for some constant b.
Compute a continued fraction representation of q.
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- For approximating the integral of f(x) over [a,b],
derive the formula for the composite trapezoidal rule:
n-1
h --- b-a
T(h) = -(f(a) + f(b)) + h > f(a+kh), h = ---
2 --- n
k=1
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| /15 |
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- Consider the values in the table below:
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| x | 2.00000 | 2.12500 | 2.25000 | 2.37500 | 2.50000 | 2.62500 | 2.75000 | 2.87500 | 3.00000 |
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|f(x)| 7.38906 | 8.37290 | 9.48773 | 10.7510 | 12.1825 | 13.8046 | 15.6426 | 17.7254 | 20.0855 |
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Perform all your calculations with six significant decimal places.
- To approximate f'(2.0), compute
forward differences Delta f(2.0,h),
for h=1,.5,.25,.125.
- Apply extrapolation using the values for Delta f(2.0,h).
- How accurate is your final approximation for f'(2.0)?
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- Consider the quadrature rule
h
/
| f(x) dx = w0 f(0) + w1 f(h/3) + w2 f(2h/3) + w3 f(h), h > 0.
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0
- Set up the system of equations in the weights w0, w1,
w2, and w3 to be satisfied for the
highest possible algebraic degree of accuracy.
Do NOT solve this system.
- What is the algebraic degree of accuracy attained by this rule?
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